Question: Consider a random walker who moves on the integers 0, 1, . . . , N, moving one step to the right with probability p

Consider a random walker who moves on the integers 0, 1, . . . , N, moving one step to the right with probability p and one step to the left with probability q = 1 − p. If the walker ever reaches 0 or N he stays there. (This is the Gambler€™s Ruin problem of Exercise 23.) If p = q show that the function f(i) = I
is a harmonic function (see Exercise 27), and if p 6= q then
f(i) = (q/ p)i
is a harmonic function. Use this and the result of Exercise 27 to show that the probability biN of being absorbed in state N starting in state i is
For an alternative derivation of these results see Exercise 24.

if p = q, ('-1 N-T if p #q. biN

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